Objectives Introduce standard transformations Introduce standard transformations Derive homogeneous coordinate transformation matrices Derive homogeneous coordinate transformation matrices Learn to build arbitrary transformation matrices from simple transformations Learn to build arbitrary transformation matrices from simple transformations CSC461: Lecture 15 Transformations

Moving the Camera In OpenGL default setting, the camera and world frames coincide with the camera pointing in the negative Z direction In OpenGL default setting, the camera and world frames coincide with the camera pointing in the negative Z direction Two ways to move objects in the front of camera: move objects or move the camera Two ways to move objects in the front of camera: move objects or move the camera M = Move camera

Model-View Matrix Fix the camera frame Fix the camera frame Model-view matrix points the world frame relative to the camera frame Model-view matrix points the world frame relative to the camera frame The model-view matrix The model-view matrix moves a point (x, y, z) in the world frame to the point (x, y, z-d) in the camera frame In OpenGL, set a model-view matrix by sending an array of 16 elements to glLoadMatrix In OpenGL, set a model-view matrix by sending an array of 16 elements to glLoadMatrix M =

A transformation maps points to other points and/or vectors to other vectors A transformation maps points to other points and/or vectors to other vectors General Transformations Q=T(P) v=T(u)

Affine Transformations Line preserving Line preserving Characteristic of many physically important transformations Characteristic of many physically important transformations Rigid body transformations: rotation, translation Rigid body transformations: rotation, translation Scaling, shear Scaling, shear Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints

Pipeline Implementation transformationrasterizer u v u v T T(u) T(v) T(u) T(v) vertices pixels frame buffer (from application program)

Notation We will be working with both coordinate-free representations of transformations and representations within a particular frame P,Q, R: points in an affine space P,Q, R: points in an affine space u, v, w: vectors in an affine space u, v, w: vectors in an affine space , ,  : scalars , ,  : scalars p, q, r: representations of points p, q, r: representations of points -array of 4 scalars in homogeneous coordinates u, v, w: representations of points u, v, w: representations of points -array of 4 scalars in homogeneous coordinates

Translation Move (translate, displace) a point to a new location Move (translate, displace) a point to a new location Displacement determined by a vector d Displacement determined by a vector d Three degrees of freedom Three degrees of freedom P’=P+d P’=P+d Although we can move a point to a new location in infinite ways, when we move many points there is usually only one way Although we can move a point to a new location in infinite ways, when we move many points there is usually only one way object translation: every point displaced by same vector P P’

Translation Using Representations Using the homogeneous coordinate representation in some frame Using the homogeneous coordinate representation in some frame p=[ x y z 1] T p=[ x y z 1] T p’=[x’ y’ z’ 1] T p’=[x’ y’ z’ 1] T d=[d x d y d z 0] T d=[d x d y d z 0] T Hence p’ = p + d or Hence p’ = p + d or x’=x+d x x’=x+d x y’=y+d y y’=y+d y z’=z+d z z’=z+d z note that this expression is in four dimensions and expresses that point = vector + point

Translation Matrix We can also express translation using a 4 x 4 matrix T in homogeneous coordinates We can also express translation using a 4 x 4 matrix T in homogeneous coordinates p’=Tp where T = T(d x, d y, d z ) = This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

Rotation Rotate objects around a line and an angle Rotate objects around a line and an angle Rotate around x-, y-, and z-axis Rotate around x-, y-, and z-axis

Rotation about the z axis Rotation about z axis in three dimensions leaves all points with the same z Rotation about z axis in three dimensions leaves all points with the same z Equivalent to rotation in two dimensions in planes of constant z Equivalent to rotation in two dimensions in planes of constant z or in homogeneous coordinates or in homogeneous coordinates p’=R z (  )p p’=R z (  )p x’=x cos  –y sin  y’=x sin  + y cos  z’=z x = r cos  y = r sin  x’ = r cos (  y’ = r sin ( 

Rotation Matrix x’=x cos θ –y sin θ y’=x sin θ + y cos θ z’=z R = R z (  ) =

Rotation about x and y axes Same argument as for rotation about z axis Same argument as for rotation about z axis For rotation about x axis, x is unchanged For rotation about x axis, x is unchanged For rotation about y axis, y is unchanged For rotation about y axis, y is unchanged R = R x (  ) = R = R y (  ) =

Scaling   Expand or contract along each axis (fixed point of origin) x’=s x x y’=s y x z’=s z x p’=Sp Matrix S = S(s x, s y, s z ) =

Reflection corresponds to negative scale factors corresponds to negative scale factors original s x = -1 s y = 1 s x = -1 s y = -1s x = 1 s y = -1

Inverses Although we could compute inverse matrices by general formulas, we can use simple geometric observations Although we could compute inverse matrices by general formulas, we can use simple geometric observations Translation: T -1 (d x, d y, d z ) = T(-d x, -d y, -d z ) Translation: T -1 (d x, d y, d z ) = T(-d x, -d y, -d z ) Rotation: R -1 (  ) = R(-  ) Rotation: R -1 (  ) = R(-  ) Holds for any rotation matrix Holds for any rotation matrix Note that since cos(-  ) = cos(  ) and sin(-  )=- sin(  ) Note that since cos(-  ) = cos(  ) and sin(-  )=- sin(  ) R -1 (  ) = R T (  ) Scaling: S -1 (s x, s y, s z ) = S(1/s x, 1/s y, 1/s z ) Scaling: S -1 (s x, s y, s z ) = S(1/s x, 1/s y, 1/s z )

Concatenation We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices We can form arbitrary affine transformation matrices by multiplying together rotation, translation, and scaling matrices Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p The difficult part is how to form a desired transformation from the specifications in the application The difficult part is how to form a desired transformation from the specifications in the application

Order of Transformations Note that matrix on the right is the first applied Note that matrix on the right is the first applied Mathematically, the following are equivalent Mathematically, the following are equivalent p’ = ABCp = A(B(Cp)) p’ = ABCp = A(B(Cp)) Note many references use column matrices to present points. In terms of column matrices Note many references use column matrices to present points. In terms of column matrices p T ’ = p T C T B T A T p T ’ = p T C T B T A T

General Rotation About the Origin A rotation by q about an arbitrary axis can be decomposed into the concatenation of rotations about the x, y, and z axes x z y v R(  ) = R z (  z ) R y (  y ) R x (  x )  x  y  z are called the Euler angles Note that rotations do not commute We can use rotations in another order but with different angles

Rotation About a Fixed Point other than the Origin Move fixed point to origin -- translate Move fixed point to origin -- translate Rotate Rotate Move fixed point back – translate back Move fixed point back – translate back M = T(-p f ) R(  ) T(p f ) M = T(-p f ) R(  ) T(p f )

Instancing In modeling, we often start with a simple object centered at the origin, oriented with the axis, and at a standard size In modeling, we often start with a simple object centered at the origin, oriented with the axis, and at a standard size We apply an instance transformation to its vertices to We apply an instance transformation to its vertices toScale Orient -- rotate Locate -- translate

Shear Helpful to add one more basic transformation Helpful to add one more basic transformation Equivalent to pulling faces in opposite directions Equivalent to pulling faces in opposite directions

Shear Matrix Consider simple shear along x axis x’ = x + y cot  y’ = y z’ = z H(  ) =